#92808

order	512
nilpotency class	3
isoclinism	isoclinic class of #7035
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#95
commutator	C₂²×C₄	C₄×C₈
Frattini, cosocle	C₂³×C₄²	C₂²
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄² ▷ C₂⁴ ▷ C₁

1066 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 1066 conjugacy classes of subgroups — too many to draw. Only its 186 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 32 points:

(1,2)(3,5)(4,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,23,24,21)(20,22)(25,26,27,29,28,30,31,32);(2,5,7,6)(9,11)(10,13,14,15)(12,16)(17,19)(18,21)(20,24)(22,23)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fh,d^2=i,b^a=bc,c^a=cf,e^a=efgh,c^b=cg,d^b=dh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267800, #72267802, #72267845
maximal normal subgroup of	#72267800, #72267802, #72267845
maximal quotient of	#72237309, #72237312, #72267815

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4);